Abstract: 
We consider the families of all subspaces of size $\omega
_1$ of $2^{\omega _1}$ (or of a compact zero-dimensional space
$X$ of weight $\omega _1$ in general) which are normal, have the
Lindel\accent"7F of property or are closed under limits of
convergent $\omega _1$-sequences. Various relations among these
families modulo the club filter in $[X]^{\omega _1}$ are shown
to be consistently possible. One of the main tools is dealing
with a subspace of the form $X\cap M$ for an elementary submodel
$M$ of size $\omega _1$. Various results with this flavor are
obtained. Another tool used is forcing and in this case various
preservation or nonpreservation results of topological and
combinatorial properties are proved. In particular we prove that
there may be no c.c.c. forcing which destroys the
Lindel\accent"7F of property of compact spaces, answering a
question of Juh\'asz. Many related questions are
formulated.

1. Departamento de Matemática
   Universidade de Säo Paulo
   Caixa Postal 66281
   Säo Paulo, SP
   CEP: 05315-970, Brasil
   
2. Departamento de Matemática
   Universidade de Säo Paulo
   Caixa Postal 66281
   Säo Paulo, SP
   CEP: 05315-970, Brasil.